Let's break this problem down step by step.

Given:

• A rectangle with dimensions 7 cm × (2x - 1) cm.
• A right-angled triangle inside the rectangle with:
  • Height 3 cm.
  • Base x cm.
• The area of the shaded region is 20 cm².

Step 1: Find the area of the rectangle

$\text{Area of rectangle} = \text{length} \times \text{width}$ $= 7(2x - 1) = 14x - 7$

Step 2: Find the area of the triangle

$\text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height}$ $= \frac{1}{2} \times x \times 3 = \frac{3x}{2}$

Step 3: Express the area of the shaded region

$\text{Shaded area} = \text{Area of rectangle} - \text{Area of triangle}$ $= (14x - 7) - \frac{3x}{2}$

Step 4: Set up the quadratic equation

The problem states that the shaded area is 20 cm², so: $(14x - 7) - \frac{3x}{2} = 20$

Multiply everything by 2 to eliminate fractions: $2(14x - 7) - 3x = 40$ $28x - 14 - 3x = 40$ $25x - 14 = 40$ $25x = 54$ $x = \frac{54}{25}$

Would you like further clarification or a step-by-step explanation?

Related Questions:

1. How do you derive the equation for the area of a shaded region in general?
2. What is the significance of forming quadratic equations in geometry?
3. How do you solve quadratic equations using factoring, completing the square, or the quadratic formula?
4. How do changes in the dimensions of the rectangle or triangle affect the quadratic equation?
5. Can this method be applied to other composite shapes like circles or trapeziums?

Math Tip:

Always check the feasibility of your solution by substituting the value of $x$ back into the original expressions to verify the given condition.